Indeed, if \(z_0 \in T \setminus E\) , then \(z_0 \notin \overline{\lbrace a_n\rbrace } \)  because otherwise
the unrestricted limit of \(B\)  at \(z_0\)  would be equal to 0, which contradicts to Step 1.
Thus, there exists an open interval \(I \subset T\)  such that \(z_0 \in I\)  and \(I\)  is disjoint of the (closed) set
\(\overline{\lbrace a_n\rbrace }\) . Then by a known theorem (see [1]}, p. 68) the Blaschke product
\(B\)  is analytic at the points of \(I\) . Thus \(B\)  is continuous on \(I\)  and so it possesses unrestricted limits
at the points of \(I\) . This means that \(I \subset T \setminus E\)  and thus \(T \setminus E\)  is open.
Hence \(E\)  is closed. The proof of necessity is completed.
